The Zappa-Sz ep product of a Fell bundle by a groupoid Boyu Li - PowerPoint PPT Presentation

The Zappa-Sz´ ep product of a Fell bundle by a groupoid Boyu Li University of Victoria November 4th, 2020 Joint work with Anna Duwenig Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 1 / 22

Background and Motivation Let G, H be two groups. Recall the semi-direct product encodes an H -action on G : ( h, x ) ∈ H × G �→ h · x ∈ G. Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 2 / 22

Background and Motivation Let G, H be two groups. Recall the semi-direct product encodes an H -action on G : ( h, x ) ∈ H × G �→ h · x ∈ G. The semi-direct product group is defined as: G ⋊ H = { ( x, h ) : x ∈ G, h ∈ H } , with multiplication and inverse: ( x, h )( y, k ) = ( x ( h · y ) , hk ) , ( x, h ) − 1 = ( h − 1 · x, h − 1 ) . Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 2 / 22

Background and Motivation Let G, H be two groups. Now, the Zappa-Sz´ ep product of G and H encodes an additional “ G -action on H ” (with some compatibility conditions with the H -action on G ): ( h, x ) ∈ H × G �→ h | x ∈ H. Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 3 / 22

Background and Motivation Let G, H be two groups. Now, the Zappa-Sz´ ep product of G and H encodes an additional “ G -action on H ” (with some compatibility conditions with the H -action on G ): ( h, x ) ∈ H × G �→ h | x ∈ H. The (external) Zappa-Sz´ ep product group is defined as: G ⊲ ⊳ H = { ( x, h ) : x ∈ G, h ∈ H } , with multiplication and inverse: ( x, h )( y, k ) = ( x ( h · y ) , h | y k ) , ( x, h ) − 1 = ( h − 1 · x, h − 1 | x − 1 ) . Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 3 / 22

Background and Motivation Let G, H be two groups. Now, the Zappa-Sz´ ep product of G and H encodes an additional “ G -action on H ” (with some compatibility conditions with the H -action on G ): ( h, x ) ∈ H × G �→ h | x ∈ H. The (external) Zappa-Sz´ ep product group is defined as: G ⊲ ⊳ H = { ( x, h ) : x ∈ G, h ∈ H } , with multiplication and inverse: ( x, h )( y, k ) = ( x ( h · y ) , h | y k ) , ( x, h ) − 1 = ( h − 1 · x, h − 1 | x − 1 ) . Note that when the G -restriction map is trivial (that is h | x = h for all h ∈ H, x ∈ G ), this coincides with the semi-direct product. Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 3 / 22

Background and Motivation Let G , H be two ´ etale groupoids. Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 4 / 22

Background and Motivation Let G , H be two ´ etale groupoids. We say they are matching if G (0) = H (0) and there exists continuous H -action and G -restriction maps: ( h, x ) �→ h · x ∈ G , s ( h ) = r ( x ) , ( h, x ) �→ h | x ∈ H , s ( h ) = r ( x ) , Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 4 / 22

Background and Motivation Let G , H be two ´ etale groupoids. We say they are matching if G (0) = H (0) and there exists continuous H -action and G -restriction maps: ( h, x ) �→ h · x ∈ G , s ( h ) = r ( x ) , ( h, x ) �→ h | x ∈ H , s ( h ) = r ( x ) , such that: ( ZS 1) ( h 1 h 2 ) · x = h 1 · ( h 2 · x ) ( ZS 2) h | xy = ( h | x ) | y ( ZS 3) r G ( x ) · x = x ( ZS 4) h | s H ( h ) = h ( ZS 5) r G ( h · x ) = r H ( h ) ( ZS 6) s H ( h | x ) = s G ( x ) ( ZS 7) h · ( xy ) = ( h · x )( h | x · y ) ( ZS 8) ( hk ) | x = h | k · x k | x ( ZS 9) s G ( h · x ) = r H ( h | x ) Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 4 / 22

Background and Motivation The (external) Zappa-Sz´ ep product groupoid is defined as G ⊲ ⊳ H = { ( x, h ) : x ∈ G , h ∈ H , r ( h ) = s ( x ) } , with multiplication and inverse: ( x, h )( y, g ) = ( x ( h · y ) , h | y g ) , ( x, h ) − 1 = ( h − 1 · x − 1 , h − 1 | x − 1 ) . Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 5 / 22

Background and Motivation The (external) Zappa-Sz´ ep product groupoid is defined as G ⊲ ⊳ H = { ( x, h ) : x ∈ G , h ∈ H , r ( h ) = s ( x ) } , with multiplication and inverse: ( x, h )( y, g ) = ( x ( h · y ) , h | y g ) , ( x, h ) − 1 = ( h − 1 · x − 1 , h − 1 | x − 1 ) . Theorem (Brownlowe, Pask, Ramagge, Robertson, Whittaker, 2017) If G , H are mathcing groupoids, then G ⊲ ⊳ H is ´ etale if and only if G and H are both ´ etale . Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 5 / 22

Background and Motivation Now in the realm of operator algebra, semi-direct product is related to the crossed product: In its simplest form, let ( A , H, α ) be a C ∗ -dynamical system. The (discrete) group H act on a C ∗ -algebra A by a ∗ -automorphic action α . One may form the algebraic crossed product: A ⋊ alg α H := { ( a, g ) : a ∈ A , g ∈ H } . We can put a ∗ -algebra structure by ( a, g )( b, h ) = ( aα g ( b ) , gh ) , ( a, g ) ∗ = ( α g − 1 ( a ∗ ) , g − 1 ) . Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 6 / 22

Background and Motivation Now in the realm of operator algebra, semi-direct product is related to the crossed product: In its simplest form, let ( A , H, α ) be a C ∗ -dynamical system. The (discrete) group H act on a C ∗ -algebra A by a ∗ -automorphic action α . One may form the algebraic crossed product: A ⋊ alg α H := { ( a, g ) : a ∈ A , g ∈ H } . We can put a ∗ -algebra structure by ( a, g )( b, h ) = ( aα g ( b ) , gh ) , ( a, g ) ∗ = ( α g − 1 ( a ∗ ) , g − 1 ) . Question What is a Zappa-Sz´ ep analogue of this? Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 6 / 22

Background and Motivation Question What is a Zappa-Sz´ ep analogue of this? Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 7 / 22

Background and Motivation Question What is a Zappa-Sz´ ep analogue of this? Several recent studies on very specific examples of Zappa-Sz´ ep type operator algebras: C ∗ -algebra of self-similar groups C ∗ -algebra of self-similar graphs and k -graphs Groupoid C ∗ -algebra of the Zappa-Sz´ ep groupoids Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 7 / 22

Background and Motivation Question What is a Zappa-Sz´ ep analogue of this? Several recent studies on very specific examples of Zappa-Sz´ ep type operator algebras: C ∗ -algebra of self-similar groups C ∗ -algebra of self-similar graphs and k -graphs Groupoid C ∗ -algebra of the Zappa-Sz´ ep groupoids To build a general framework, there are two key ingredients: The C ∗ -algebra A has to “act” on the group H in a non-trivial way. This forces some kind of grading on A . We also need to define a notion of the ∗ -automorphic action α , that is compatible with the Zappa-Sz´ ep structure. Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 7 / 22

Fell Bundle A Fell bundle provides a grading: Definition A Fell bundle B = ( B, p ) over a groupoid G is a upper semicontinuous Banach bundle equipped with continuous multiplication and involution such that For each ( x, y ) ∈ G (2) , B x · B y ⊂ B xy . The multiplication is bilinear and associative. For any b, c ∈ B , � b · c � ≤ � b �� c � . For any x ∈ G , B ∗ x ⊂ B x − 1 . The involution is conjugate linear. For any b, c ∈ B , ( bc ) ∗ = c ∗ b ∗ and b ∗∗ = b . For any b ∈ B , � b ∗ b � = � b � 2 = � b ∗ � 2 . For any b ∈ B , b ∗ b ≥ 0 as an element in B s ( p ( b )) . Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 8 / 22

Compatible Action We now need a compatible action: Definition Let B = ( B, p ) be a Fell bundle over an ´ etale groupoid G , and let H be a matching ´ etale groupoid. A ( G , H ) -compatible H -action on B is a continuous map: β : ( h, b ) �→ β h ( b ) , s ( h ) = r ( p ( b )) , satisfying: β h is a linear map from B x to B h · x for all s ( h ) = r ( x ) . For any ( g, h ) ∈ H (2) , β g ◦ β h = β gh . For any u ∈ H (0) , β u is the identity map. For any bc ∈ B and r ( p ( b )) = s ( h ) , β h ( bc ) = β h ( b ) β h | p ( b ) ( c ) . For any b ∈ B with r ( p ( b )) = s ( h ) , β h ( b ) ∗ = β h | p ( b ) ( b ∗ ) . Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 9 / 22

Compatible Action The maps { β h } are not ∗ -automorphic as in the semi-crossed product. However, they do enjoy some nice properties: Proposition For each h ∈ H , β h : B s ( h ) → B r ( h ) is an injective ∗ -isomorphism of C ∗ -algebras. Proposition For each h ∈ H and x ∈ G with s ( h ) = r ( x ) , β h : B x → B h · x is isometric. Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle Nov. 4th, 2020 10 / 22